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Update completed (POTE -> CTE; review comma lists and GPV sequences)
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{{Mapping|legend=1| 1 0 2 | 0 5 1 }}
{{Mapping|legend=1| 1 0 2 | 0 5 1 }}


: mapping generators: ~2, ~5/4
: Mapping generators: ~2, ~5/4


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~5/4 = 380.4994
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~5/4 = 380.4994
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=== Overview to extensions ===
=== Overview to extensions ===
Apart from magic, we also consider other extensions. The second comma of the [[Normal lists #Normal interval list|normal comma list]] defines which 7-limit family member we are looking at. [[875/864]], the keemic comma, gives septimal magic, and [[525/512]], Avicenna's enharmonic diesis, gives his annoying brother muggles. Both use the major third as a generator.  
Apart from magic, we also consider other extensions. The second comma of the [[Normal lists #Normal interval list|normal comma list]] defines which 7-limit family member we are looking at. [[875/864]], the keemic comma, gives septimal magic, and [[525/512]], Avicenna's enharmonic diesis, gives his annoying brother muggles. Both use the major third as a generator, as well as low-accuracy extensions including darkstone and brightstone.  


Weak extensions considered below are hocum, trismegistus, quadrimage, and warlock. Discussed elsewhere are
Weak extensions considered below are hocum, trismegistus, quadrimage, quinmage and warlock. Discussed elsewhere are
* ''[[Astrology]]'' → [[Jubilismic clan #Astrology|Jubilismic clan]]
* ''[[Astrology]]'' → [[Jubilismic clan #Astrology|Jubilismic clan]]
* ''[[Spell]]'' → [[Hemimean clan #Spell|Hemimean clan]]
* ''[[Spell]]'' → [[Hemimean clan #Spell|Hemimean clan]]
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{{Mapping|legend=1| 1 0 2 -1 | 0 5 1 12 }}
{{Mapping|legend=1| 1 0 2 -1 | 0 5 1 12 }}


: mapping generators: ~2, ~5/4
: Mapping generators: ~2, ~5/4


{{Multival|legend=1| 5 1 12 -10 5 25 }}
{{Multival|legend=1| 5 1 12 -10 5 25 }}
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== Darkstone ==
== Darkstone ==
''Darkstone'' (16 & 19d) is a low-accuacy temperament which tempers out 36/35 and 1875/1792. It makes the major third and the fifth even flatter than those of muggles. In [http://www.tonalsoft.com/enc/m/magic.aspx ''Encyclopedia of Microtonal Music Theory'', Tonalsoft], this temperament is given a name ''witch''.
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


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== Brightstone ==
== Brightstone ==
{{see also| Archytas clan }}
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


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[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~5/4 = 381.955
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~5/4 = 381.955


{{Optimal ET sequence|legend=1| 19d, 22 }}
{{Optimal ET sequence|legend=1| 3, …, 19d, 22 }}


[[Badness]]: 0.088072
[[Badness]]: 0.088072
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Optimal tuning (CTE): ~2 = 1\1, ~5/4 = 381.790
Optimal tuning (CTE): ~2 = 1\1, ~5/4 = 381.790


Optimal ET sequence: {{Optimal ET sequence| 19d, 22 }}
Optimal ET sequence: {{Optimal ET sequence| 3, …, 19d, 22 }}


Badness: 0.047379
Badness: 0.047379
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 64/63, 65/63, 100/99, 169/165
Mapping: {{mapping| 1 0 2 6 6 4 | 0 5 1 -10 -8 -1 }}
Optimal tuning (CTE): ~2 = 1\1, ~5/4 = 381.732
Optimal ET sequence: {{Optimal ET sequence| 3, …, 19d, 22 }}
Badness: 0.039703


== Hocum ==
== Hocum ==
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{{Mapping|legend=1| 1 5 3 -3 | 0 -10 -2 17 }}
{{Mapping|legend=1| 1 5 3 -3 | 0 -10 -2 17 }}


: mapping generators: ~2, ~63/50
: Mapping generators: ~2, ~63/50


{{Multival|legend=1| 10 2 -17 -20 -55 -45 }}
{{Multival|legend=1| 10 2 -17 -20 -55 -45 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~63/50 = 410.108
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~63/50 = 409.836


{{Optimal ET sequence|legend=1| 38, 41, 161c, 202c, 243c, 284c }}
{{Optimal ET sequence|legend=1| 3, …, 35d, 38, 41, 202cc, 243cc, 284cc }}


[[Badness]]: 0.107115
[[Badness]]: 0.107115
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{{Mapping|legend=1| 1 10 4 0 | 0 -15 -3 5 }}
{{Mapping|legend=1| 1 10 4 0 | 0 -15 -3 5 }}


: mapping generators: ~2, ~147/100
: Mapping generators: ~2, ~147/100


{{Multival|legend=1| 15 3 -5 -30 -50 -20 }}
{{Multival|legend=1| 15 3 -5 -30 -50 -20 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~147/100 = 673.290
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~147/100 = 673.187


{{Optimal ET sequence|legend=1| 16, 25, 41, 139c, 180c, 221c, 262c }}
{{Optimal ET sequence|legend=1| 16, 25, 41 }}


[[Badness]]: 0.098334
[[Badness]]: 0.098334
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Mapping: {{mapping| 1 10 4 0 13 | 0 -15 -3 5 -17 }}
Mapping: {{mapping| 1 10 4 0 13 | 0 -15 -3 5 -17 }}


Optimal tuning (POTE): ~2 = 1\1, ~22/15 = 673.340
Optimal tuning (CTE): ~2 = 1\1, ~22/15 = 673.241


{{Optimal ET sequence|legend=1| 16, 25e, 41, 98c, 139c, 180c }}
Optimal ET sequence: {{Optimal ET sequence| 16, 25e, 41, 221cc, 262ccde }}


Badness: 0.045623
Badness: 0.045623
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Mapping: {{mapping| 1 10 4 0 13 11 | 0 -15 -3 5 -17 -13 }}
Mapping: {{mapping| 1 10 4 0 13 11 | 0 -15 -3 5 -17 -13 }}


Optimal tuning (POTE): ~2 = 1\1, ~22/15 = 673.359
Optimal tuning (CTE): ~2 = 1\1, ~22/15 = 673.294


{{Optimal ET sequence|legend=1| 16, 25e, 41, 98c, 139cf }}
Optimal ET sequence: {{Optimal ET sequence| 16, 25e, 41, 139cf, 180cf, 221ccf }}


Badness: 0.033081
Badness: 0.033081
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{{Mapping|legend=1| 1 5 3 4 | 0 -20 -4 -7 }}
{{Mapping|legend=1| 1 5 3 4 | 0 -20 -4 -7 }}


: mapping generators: ~2, ~28/25
: Mapping generators: ~2, ~28/25


{{Multival|legend=1| 20 4 7 -40 -45 5 }}
{{Multival|legend=1| 20 4 7 -40 -45 5 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/25 = 204.987
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~28/25 = 204.860


{{Optimal ET sequence|legend=1| 35, 41, 158cd, 199ccd, 240ccd, 281ccd }}
{{Optimal ET sequence|legend=1| 6, …, 29b, 35, 41 }}


[[Badness]]: 0.127422
[[Badness]]: 0.127422
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Mapping: {{mapping| 1 5 3 4 5 | 0 -20 -4 -7 -9 }}
Mapping: {{mapping| 1 5 3 4 5 | 0 -20 -4 -7 -9 }}


Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 204.956
Optimal tuning (CTE): ~2 = 1\1, ~28/25 = 204.881


{{Optimal ET sequence|legend=1| 35, 41 }}
Optimal ET sequence: {{Optimal ET sequence| 6, …, 29b, 35, 41 }}


Badness: 0.061572
Badness: 0.061572
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Mapping: {{mapping| 1 5 3 4 5 9 | 0 -20 -4 -7 -9 -31 }}
Mapping: {{mapping| 1 5 3 4 5 9 | 0 -20 -4 -7 -9 -31 }}


Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 205.028
Optimal tuning (CTE): ~2 = 1\1, ~28/25 = 204.956


{{Optimal ET sequence|legend=1| 35f, 41, 117c }}
Optimal ET sequence: {{Optimal ET sequence| 35f, 41, 199ccdef, 240ccdef, 281ccdeff }}


Badness: 0.044047
Badness: 0.044047
== Quinmage ==
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 3125/3072, 16875/16807
{{Mapping|legend=1| 1 -10 0 -6 | 0 25 5 19 }}
{{Multival|legend=1| 25 5 19 -50 -40 30 }}
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~48/35 = 556.123
{{Optimal ET sequence|legend=1| 13b, 28b, 41 }}
[[Badness]]: 0.194548
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 385/384, 625/616, 2401/2376
Mapping: {{mapping| 1 -10 0 -6 3 | 0 25 5 19 1 }}
Optimal tuning (CTE): ~2 = 1\1, ~11/8 = 556.122
Optimal ET sequence: {{Optimal ET sequence| 13b, 28b, 41 }}
Badness: 0.101724
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 196/195, 364/363, 385/384, 625/616
Mapping: {{mapping| 1 -10 0 -6 3 0 | 0 25 5 19 1 8 }}
Optimal tuning (CTE): ~2 = 1\1, ~11/8 = 556.106
Optimal ET sequence: {{Optimal ET sequence| 13b, 28b, 41 }}
Badness: 0.067742


== Warlock ==
== Warlock ==
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{{Mapping|legend=1| 5 0 10 14 | 0 5 1 0 }}
{{Mapping|legend=1| 5 0 10 14 | 0 5 1 0 }}


: mapping generators: ~8/7, ~5/4
: Mapping generators: ~8/7, ~5/4


[[Optimal tuning]] ([[POTE]]): ~8/7 = 1\5, ~5/4 = 379.7131
[[Optimal tuning]] ([[CTE]]): ~8/7 = 1\5, ~5/4 = 380.499


{{Optimal ET sequence|legend=1| 25, 35, 60 }}
{{Optimal ET sequence|legend=1| 25, 35, 60 }}
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